The Distance Squared Function on a Planar Curve

Given a regular unit speed curve
in
, for a fixed point
in
, define,

(21)

We are interested in finding the points on the curve which have either local minimal or local maximal distances to
. That is,
for a fixed point
, we are trying to find those
s, such that
and either
or
.

has a non-singular submatrix consisting of the last two columns if
, or a non-sinuglar submatrix consisting of the first column and either the seond or the third column
(
and
can not be both zero) if
.

When projecting back to
, we have the envelope of the normals of curve, which is just the evolute of the curve, because from equation 25, we have,

and

(27)

The evolute is therefore smooth so long as
(See Section 3). Otherwise, we have the regression point,

Now back to our minimal or maximal distance problem, we have the following local result,

For a fixed point
, the local minimal or maximal distance point has to be the curve point, say
, with the normal passing through
. See equation (24).

is a local minimal distance point if
, i.e.,
, or
is closer to it than the curvature center.

is a local maximal distance point if
, i.e.,
is farther to it than the curvature center.

if
is not a vertex, and
is its curvature center, i.e.
, but
, then
is neither a local minimal distance point nor a local maximal one.

is a local minimal distance point, if it is a vertex of maximal curvatur, and
is its curvature center. i.e.
, and
, but
.

is a local maximal distance point, if it is a vertex of minimal curvatur, and
is its curvature center. i.e.
, and
, but
.

In the neighborhood of curvature center of vertex of maximal curvature, we have the following multi-local result(Figure 3),

For any point
, outside the folded area, there is only one minimal distance point. (
-singularity)

For any point
, inside the folded area, there are two minimal distance points and one maximal point. (all
-singularity)

For any point
, on the fold but not the cusp, there is one minimal distance point.(
-singularity, and there is another curve point of
singularity).

For
on the cusp, there is one minimal distance point. (
singularity).

The evolute divides the
control space into 2 regions, with 1 and 3 critical points respectively. When moving the point
from the 1 critical point (which is minimal) region into
the 3 critical point region, the change of minimal point on the curve is gradual. If we keep moving point
all the way through the 3 critical point region and crossing out into the other side
of the 1 critical point region, there is a sudden change of minimal point on the curve, which is called catastrophe.

Figure:
Evolute of the parabola
. It has an ordinary cusp, which is also the local structure of any versal 2-unfolding of
singularity